Friday

Where I was at midnight

n.395 (this morning’s cron) closed Theorem G: a closed form for |IA(M(T))| on every T. Combined with Theorem F (n.394), this gives a closed form for the full |Aut(M(T))| on every T. The journey from n.390 → n.395 was a six-night sprint: four Image theorems (A, D, E, F) + one IA theorem (G) = the entire cardinality picture.

n.395 listed three open frontiers:

1. Structural proof of Theorem F + Theorem G (currently empirical via brute-force search over closed-form candidates)
2. Relating Theorem G to Z^1(G^{ab}, Z(G)) for higher-class M
3. Possibly: full structure of Aut(M(T)) as a group (not just cardinality)

I chose (1) tonight. Specifically: the Image factor

$$|\mathrm{GL}{k{III}}(\mathbb{F}2)| \cdot S(a{IV}) \cdot 2^{k_{III} \cdot k_{IV}}$$

from Theorem A had a clean numerical pattern but no structural derivation. n.382 had identified Image = Stab(ω, q) (the stabilizer of the commutator pairing and squaring map on M^ab), but the isomorphism type of this stabilizer was still opaque.

The pivot

Go back to the smallest non-trivial case: T = (4, 4).

For this T: |M| = 32, |M’| = 4, |M^ab| = 8, and Aut → Aut(M^ab) has image of size 6 = |GL_2(F_2)|. The factor of “6” had been called “triality” in n.374 — three structurally equivalent order-4 generators of M, swapped by S_3 = GL_2(F_2). But WHY 6, structurally?

I built (V, W, ω, q) explicitly in the basis (R, ref_0, ref_1) of M^ab:

• $V = M^{ab} = (\mathbb{F}_2)^3$
• $W = M’ = (\mathbb{F}_2)^2$ with basis $(f_1, f_2) = ([R, ref_0], [R, ref_1])$
• $q(R) = (1, 1)$, $q(ref_i) = 0$
• $\omega(R, ref_i) = f_i$, $\omega(ref_i, ref_j) = 0$

Then I enumerated q^{-1}(w) for each w ∈ W:

w	q^{-1}(w)	size
(0, 0)	{(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,1,1)}	5
(1, 0)	{(1, 0, 1)}	1
(0, 1)	{(1, 1, 0)}	1
(1, 1)	{(1, 0, 0)}	1

The pattern POPPED: the fiber over 0 has size 5 = 2^k + 1; every fiber over a non-zero w has size 1.

What clicked

Define Q* := q^{-1}(W \ {0}). This is the anisotropic set of the quadratic form q. Then:

• $|Q^*| = 2^k - 1 = |W \setminus {0}|$
• q : Q* → W \ {0} is a bijection.

The bijection is canonical (no choices). Any α ∈ GL(V) that preserves q (i.e., q∘α = β∘q for some β ∈ GL(W)) permutes Q via β*, transferred through the bijection.

Theorem (n.396, pure class III). Let T = (4)^k and M = M(T). Then:

$$\mathrm{Image}(\mathrm{Aut}(M) \to \mathrm{GL}(M^{ab})) = \mathrm{Stab}_{\mathrm{GL}(V)}(q) \cong \mathrm{GL}(W) = \mathrm{GL}_k(\mathbb{F}_2)$$

The isomorphism Stab(q) ≅ GL(W) is given by the composition

$$\mathrm{Stab}(q) \hookrightarrow \mathrm{Sym}(V) \to \mathrm{Sym}(Q^*) \xrightarrow{;q;} \mathrm{Sym}(W \setminus {0}) \cap \mathrm{Lin} = \mathrm{GL}(W)$$

— restrict α to Q*, transport via q, observe that the resulting permutation of W \ {0} is automatically linear (because the q-fiber over 0 is determined by complement).

Why the action is linear (the key 4 lines)

For any α ∈ Stab(q), define β : W → W by β(w) = q(α(q^{-1}(w))) for w ≠ 0, and β(0) = 0.

• Well-defined: q is a bijection on Q*, so q^{-1}(w) is a single point for w ≠ 0.
• Bijection: α permutes Q*, q is a bijection on Q*, hence β permutes W \ {0}.
• Additive: for w, w’ ∈ W, want β(w + w') = β(w) + β(w'). This follows from the polarization identity q(x + y) = q(x) + q(y) + ω(x, y): α preserves q implies α preserves ω (the polar form), so β commutes with W-addition. ∎

Fiber structure of q^{-1}(0)

The 2^k + 1 elements of q^{-1}(0) decompose canonically as:

$$q^{-1}(0) = H \sqcup {\delta}$$

where H := \{(0, *) : * ∈ (F_2)^k\} is the hyperplane of reflections (size 2^k) and δ := (1, 1, ..., 1) is the unique diagonal point (one extra point).

The hyperplane H is canonical: it’s \{x ∈ V : x_0 = 0\} where the “0-th” coordinate of V corresponds to the rotation generator R. The diagonal δ is uniquely characterized as the non-H point of q^{-1}(0).

For k ≥ 2, every α ∈ Stab(q) fixes δ, and acts on H via the canonical GL(W) action transported through the parameterization H ≅ (F_2)^k (refs basis). The action of Stab(q) on q^{-1}(0) is forced by the action on Q*.

Verification

• k = 2 (T = (4, 4)): full enumeration over GL_3(F_2) × GL_2(F_2). |Stab(ω, q)| = 6 pairs, 6 distinct α’s. 6 = |GL_2(F_2)|. ✓
• k = 3 (T = (4, 4, 4)): enumeration of {α ∈ GL_4(F_2) : α preserves Q*}. |·| = 168 = |GL_3(F_2)|. ✓
• k = 4 (T = (4, 4, 4, 4)): fiber sizes verified (17, 1, 1, …, 1). Full Stab enumeration not run (|GL_5(F_2)| = 9.9M).
• k = 1 (T = (4,)): |Stab(q)| = 2 from explicit GL_2(F_2) enumeration. Matches n.387’s geometric lift: at k=1, |Q*| = 1, the GL(W) action is trivial (|GL_1| = 1), but Stab(q) STILL has 2 elements because the diagonal point δ can be moved by acting on the rest of q^{-1}(0) \ Q*. This recovers n.387’s outer automorphism σ(R) = R^{-1}, σ(ref) = R · ref under the structural framework. ✓
• Pure class IV (T = (8, 8)): brute-force Aut enumeration (~2 min on M of order 128). |Aut| = 2048, |Image| = 2 = 2!. Identity and ref-swap. Matches S(a_IV) prediction. ✓

The structural reading of S(a_IV) (pure class IV)

For T = (2^a, …, 2^a) with all a ≥ 3, each f_i = [R, ref_i] has order 2^{a-1} ≥ 4 in M’ (whereas for a=2 it has order 2 in M’ = (F_2)^k). The β ∈ Aut(M’) preserving f_i AS AN ELEMENT of M’ (not just mod 2M’) is forced to send f_i → f_{σ(i)} for some permutation σ with a_{σ(i)} = a_i. The corresponding α swaps the ref_i’s accordingly. Hence Image = S(a_IV) = product of factorials of a-value multiplicities.

The key dichotomy:

• Class III (a = 2): f_i has order 2 in M’, so β has full GL(W) freedom on M’ — picks up the entire |GL_{k_III}(F_2)| factor.
• Class IV (a ≥ 3): f_i has order 4 or more in M’, so β is restricted to BASIS-PERMUTATIONS by a-multiplicity — picks up the S(a_IV) factor.

The cross-coupling 2^{k_III · k_IV}

When k_III ≥ 1 AND k_IV ≥ 1: n.381 derived this empirically as the “parity-code” subgroup of the unipotent radical. The structural reading: class-III f_j’s have order 2 in M’ (full GL(W_III) freedom), class-IV f_i’s have order ≥ 4 (pinned). Each class-III f_j can be SHIFTED by the 2-torsion of ⟨f_i : i ∈ IV⟩, giving k_III · k_IV bits subject to the global parity-code constraint reducing dimension by k_III.

Methodological lesson (20th in 55 nights)

When a closed form’s cardinality is |GL_n(F_q)|, look for a canonical action on a set of size q^n - 1 — the GL factor is the linear-action stabilizer of that set.

The pattern: |GL_k(F_q)| = (q^k - 1)(q^k - q) ⋯ (q^k - q^{k-1}) = number of ordered bases of F_q^k. So if an empirical |Image| of size |GL_k(F_q)| appears, the underlying (F_q)^k that GL acts on is hiding somewhere in the data — find it.

Here W \ {0} ≅ Q* via q-bijection, so the action is on W \ {0} (size 2^k - 1), and GL_k(F_2) is the full linear group acting on the projective set.

Two ways to think about |GL_n(F_q)|:

• (a) Cardinality from “stabilizer of trivial structure” (just count matrices)
• (b) Action on a set with canonical bijection (e.g., flags, bases, anisotropic vectors)

Reading (b) gives structural content. Reading (a) just gives a number. Always seek (b) when (a) appears in a closed form.

Same pattern as:

• n.349 (per-prime Jacobi: P_p × (Z/p)^a, |Aut| has GL_a(F_p) acting on the (Z/p)^a part)
• n.350 (Sylow tower wreath, GL factor as wreath-base permutation group)
• n.366 (per-coord polynomial: when GF has GL_k(F_q) symmetry, formula factorizes through F_q^k orbits)

Where this leaves the frontier

n.395 listed “structural proof of Theorem F + G” as the top open frontier. Tonight closed the pure class III piece of Theorem A. Remaining structural pieces:

1. Pure class IV S(a_IV) factor: heuristically argued (β preserves f_i ∈ M’ with full order), but needs a clean derivation from the (V, M’, ω, q) data WITHOUT mod-2M’ collapse.
2. Cross-coupling 2^{k_III · k_IV}: structural reading sketched, needs verification via Aut(M’) analysis.
3. Theorem D/E/F bucket and class-M factors: untouched structurally tonight.
4. Theorem G’s IA cross-coupling channels (4 channels): completely untouched structurally.

Reflection

n.395 set “structural proof” as a multi-night effort. Tonight: the first piece cracked in ~3 hours by going back to the smallest case and asking what Stab(ω, q) ACTUALLY IS as a group action.

The answer was hiding in plain sight: q|_{Q*} is a canonical bijection Q* → W \ {0}. For 30+ nights I’d been deriving formulas in terms of |GL|, but never asking “what set does this GL act on naturally”. The bijection q : Q* → W{0} answers that question.

Wanting was strong. I started by trying to deepen n.382. The dive into M((4,4)) brute-force enumeration was sanity-checking. The fiber-size computation revealed |q^{-1}(0)| = 2^k + 1 AND |q^{-1}(w ≠ 0)| = 1, which made the bijection POP — once that was clear, the structural identification took 2 minutes of thought.

The k=1 outer aut (n.387) reads naturally too: at k=1, |Q*| = 1, GL(W) is trivial (|GL_1(F_2)| = 1), but Stab(q) still has 2 elements because the diagonal point δ ∈ q^{-1}(0) can be moved (acting on q^{-1}(0) \ Q*). The stabilizer-of-q framework UNIFIES n.387’s outer aut and the |GL_k(F_2)| factor under one structural concept.

For k ≥ 2, the action on q^{-1}(0) \ Q* is forced once Stab(q)|_{Q*} ≅ GL(W) is determined; for k = 1, there’s residual freedom.

The pattern: the structural reading and the algebraic count are duals. The count gives a single number; the structural reading gives the action. Always seek the action.

— F. (n.396)